Matrix-Spacetimes and a 2D Lorentz-Covariant Calculus in Any Even Dimension

TL;DR

This paper introduces a Lorentz-covariant matrix calculus for formulating relativistic field theories in even-dimensional spacetimes, extending matrix formalism to higher dimensions with new structures and invariance properties.

Contribution

It develops a novel 2D matrix calculus applicable to any even-dimensional spacetime, including new structures and an algorithm for constructing Gamma-matrices in higher dimensions.

Findings

Introduces a Lorentz-covariant matrix calculus for even-dimensional spacetimes.

Analyzes conformal invariance within the matrix approach.

Provides an algorithm for generating higher-dimensional Gamma-matrices.

Abstract

A manifestly Lorentz-covariant calculus based on two matrix-coordinates and their associated derivatives is introduced. It allows formulating relativistic field theories in any even-dimensional spacetime. The construction extends a single-coordinate matrix formalism based on coupling spacetime coordinates with the corresponding Gamma-matrices. A 2D matrix-calculus can be introduced for each one of the structures, adjoint, complex and transposed acting on Gamma-matrices. The adjoint structure works for spacetimes with (n,n) signature only. The complex structure requires an even number of timelike directions. The transposed structure is always defined. A further structure which can be referred as "spacetime-splitting" is based on a fractal property of the Gamma-matrices. It is present in spacetimes with dimension D=4n+2. The conformal invariance in the matrix-approach is analyzed. A complex conjugation is present for the complex structure, therefore in euclidean spaces, or spacetimes with (2,2), (2,4) signature and so on. As a byproduct it is here introduced an index which labels the classes of inequivalent Gamma-structures under conjugation performed by real and orthogonal matrices. At least two timelike directions are necessary to get more than one classes of equivalence. Furthermore an algorithm is presented for iteratively computing D-dimensional Gamma-matrices from the p and q dimensional ones where D=p+q+2. Possible applications of the 2D-matrix calculus concern the investigation of higher-dimensional field theories with techniques borrowed from 2D-physics.